Investments

Kevin Crotty

Portfolio Statistics

Portfolios

-Portfolio are combinations of underlying assets
-Given return properties of the underlying assets, what are the return properties of their combination?

Expected Return of Portfolio of \(N\) Assets

\[ E[r_p] = \sum_{i=1}^{N} w_i E[r_i] \]

-\(w_i\) is the portfolio weight of asset \(i\)
-\(E[r_i]\) is the expected return of asset \(i\)
-The portfolio is fully invested: \(\sum_i w_i = 1\)
-Notation: \(E(r_p)=\mu_i\)

Variance of Portfolio of \(N\) Assets

\[ \text{var}[r_p] = \sum_{i=1}^{N} \sum_{j=1}^{N} w_i w_j \text{cov}[r_i,r_j] \]

-\(w_i\) is the portfolio weight of asset \(i\)
-\(\text{cov}[r_i,r_j]\) is the covariance between assets \(i\) and \(j\)
-Recall that \(\text{cov}[r_i,r_j]=\text{var}[r_i]\) and \(\text{sd}[r_i]=\sqrt{\text{var}[r_i]}\)
-Notation: \(\text{var}[r_p]=\sigma^2_p\); \(\text{cov}[r_i,r_j]=\sigma_{i,j}\); \(\text{sd}[r_p]=\sigma_p\)

Variance of Portfolio of \(N\) Assets: A Matrix View

\[ \text{var}[r_p] = \sum_{i=1}^{N} \sum_{j=1}^{N} w_i w_j \text{cov}[r_i,r_j] \]

\(w_1 w_1 \text{cov}[r_1,r_1]\) \(w_1 w_2 \text{cov}[r_1,r_2]\) \(w_1 w_3 \text{cov}[r_1,r_3]\)
\(w_2 w_1 \text{cov}[r_2,r_1]\) \(w_2 w_2 \text{cov}[r_2,r_2]\) \(w_2 w_3 \text{cov}[r_2,r_3]\)
\(w_3 w_1 \text{cov}[r_3,r_1]\) \(w_3 w_2 \text{cov}[r_3,r_2]\) \(w_3 w_3 \text{cov}[r_3,r_3]\)

Variance of Portfolio of \(N\) Assets: A Matrix View

\[ \text{var}[r_p] = \sum_{i=1}^{N} \sum_{j=1}^{N} w_i w_j \text{cov}[r_i,r_j] \]

\(w_1^2 \text{var}[r_1]\) \(w_1 w_2 \text{cov}[r_1,r_2]\) \(w_1 w_3 \text{cov}[r_1,r_3]\)
\(w_2 w_1 \text{cov}[r_2,r_1]\) \(w_2^2 \text{var}[r_2]\) \(w_2 w_3 \text{cov}[r_2,r_3]\)
\(w_3 w_1 \text{cov}[r_3,r_1]\) \(w_3 w_2 \text{cov}[r_3,r_2]\) \(w_3^2 \text{var}[r_3]\)

Variance of Portfolio of \(N\) Assets: A Matrix View

\[ \text{var}[r_p] = \sum_{i=1}^{N} w_i^2 \text{var}[r_i]+ 2 \sum_{j>i} w_i w_j \text{cov}[r_i,r_j] \]

\(w_1^2 \text{var}[r_1]\) \(w_1 w_2 \text{cov}[r_1,r_2]\) \(w_1 w_3 \text{cov}[r_1,r_3]\)
\(w_2 w_1 \text{cov}[r_2,r_1]\) \(w_2^2 \text{var}[r_2]\) \(w_2 w_3 \text{cov}[r_2,r_3]\)
\(w_3 w_1 \text{cov}[r_3,r_1]\) \(w_3 w_2 \text{cov}[r_3,r_2]\) \(w_3^2 \text{var}[r_3]\)

Example: Equal-weighted portfolio of two assets

-Expected Return

\[\begin{align} E[r_p] =& w_1 E[r_1] + w_2 E[r_2] \\ =& 0.5 E[r_1] + 0.5 E[r_2] \\ \end{align}\]

-Portfolio Variance

\[\begin{align} \text{var}[r_p] =& w_1^2 \text{var}[r_1]+ w_2^2 \text{var}[r_2]+ 2 w_1 w_2 \text{cov}[r_1,r_2] \\ =& 0.5^2 \text{var}[r_1]+ 0.5^2 \text{var}[r_2]+ 2\cdot 0.5\cdot 0.5 \text{cov}[r_1,r_2] \\ =& 0.25 \text{var}[r_1]+ 0.25 \text{var}[r_2]+ 0.5 \text{cov}[r_1,r_2] \\ \end{align}\]

Variance of Portfolio of \(N\) Assets: Matrices

\[ \text{var}[r_p] = \sum_{i=1}^{N} \sum_{j=1}^{N} w_i w_j \text{cov}[r_i,r_j] = w'Vw \]

-Portfolio weights vector:

\[w'=[w_1\, w_2\,...\,w_N]\]

-Covariance matrix of returns: \[\begin{equation*} V = \begin{bmatrix} \text{var}[r_1] & \text{cov}[r_1,r_2] & \dots & \text{cov}[r_1,r_N] \\ \text{cov}[r_2,r_1] & \text{var}[r_2] & \dots & \text{cov}[r_2,r_N] \\ \vdots & \vdots & \ddots & \vdots \\ \text{cov}[r_N,r_1] & \text{cov}[r_N,r_2] & \dots & \text{var}[r_N] \\ \end{bmatrix} \end{equation*}\]

Covariance and Correlation

-Covariance: absolute degree of co-movement between two assets
-Correlation: relative degree of co-movement between two assets

\[ \text{corr}[r_i,r_j] = \rho_{ij} = \frac{\text{cov}[r_i,r_j]}{\text{sd}[r_i]\cdot\text{sd}[r_j]} \]

-What are the possible values for \(\rho\)?

Diversification

Effects of Diversification

Claim: The variance of the return of a portfolio with many securities depends more on the covariances between the individual securities than on the variances of the individual securities.

\(w_1^2 \text{var}[r_1]\) \(w_1 w_2 \text{cov}[r_1,r_2]\) \(w_1 w_3 \text{cov}[r_1,r_3]\)
\(w_2 w_1 \text{cov}[r_2,r_1]\) \(w_2^2 \text{var}[r_2]\) \(w_2 w_3 \text{cov}[r_2,r_3]\)
\(w_3 w_1 \text{cov}[r_3,r_1]\) \(w_3 w_2 \text{cov}[r_3,r_2]\) \(w_3^2 \text{var}[r_3]\)

Now consider a five-security portfolio

How many variance terms?

\[N=5\]

How many covariance terms?

\[N^2-N = 25-5=20 \]

\(w_1^2 \text{var}[r_1]\) \(w_1 w_2 \text{cov}[r_1,r_2]\) \(w_1 w_3 \text{cov}[r_1,r_3]\) \(w_1 w_4 \text{cov}[r_1,r_4]\) \(w_1 w_5 \text{cov}[r_1,r_5]\)
\(w_2 w_1 \text{cov}[r_2,r_1]\) \(w_2^2 \text{var}[r_2]\) \(w_2 w_3 \text{cov}[r_2,r_3]\) \(w_2 w_4 \text{cov}[r_2,r_4]\) \(w_2 w_5 \text{cov}[r_2,r_5]\)
\(w_3 w_1 \text{cov}[r_3,r_1]\) \(w_3 w_2 \text{cov}[r_3,r_2]\) \(w_3^2 \text{var}[r_3]\) \(w_3 w_4 \text{cov}[r_3,r_4]\) \(w_3 w_5 \text{cov}[r_3,r_5]\)
\(w_4 w_1 \text{cov}[r_4,r_1]\) \(w_4 w_2 \text{cov}[r_4,r_2]\) \(w_4 w_3 \text{cov}[r_4,r_3]\) \(w_4^2 \text{var}[r_4]\) \(w_4 w_5 \text{cov}[r_4,r_5]\)
\(w_5 w_1 \text{cov}[r_5,r_1]\) \(w_5 w_2 \text{cov}[r_5,r_2]\) \(w_5 w_3 \text{cov}[r_5,r_3]\) \(w_5 w_4 \text{cov}[r_5,r_4]\) \(w_5^2 \text{var}[r_5]\)

Why does covariance dominate with large \(N\)?

-Consider an \(N\)-asset, equal-weighted portfolio \((w=1/N\))
-Assume all assets have the same variance \((\sigma^2_{\text{all}})\)
-Assume all pairs of assets have the same covariance \((\overline{\text{cov}})\)
-What is the variance of the portfolio?

\[\begin{align*} \text{var}(r_p) =& N\cdot \left(\frac{1}{N}\right)^2 \sigma^2_{\text{all}} + (N^2-N)\cdot \left(\frac{1}{N}\right)^2 \overline{\text{cov}} \\ =& \left(\frac{1}{N}\right) \sigma^2_{\text{all}} + \left(\frac{N-1}{N}\right) \overline{\text{cov}} \end{align*}\]

What happens to this as N gets large?

\[ \text{var}(r_p) \underset{N\rightarrow \infty}{\longrightarrow} 0\cdot\sigma^2_{\text{all}}+(1-0)\overline{\text{cov}}=\overline{\text{cov}} \]

Diversification curves

-Diversification eliminates some, but not all, of the risk of individual assets.
-In large portfolios, \(\text{var}[r_i]\)’s effectively diversified away, but not \(\text{cov}[r_i,r_j]\)’s.
-Diversifiable, non-systematic, idiosyncratic risk vs. non-diversifiable, systematic, market risk

Diversification curves

INSERT PLOTS OR WEBPAGE OF EMPIRICAL DIVERSIFICATION CURVES?

Preferences

Which return series do you prefer?

Code 
import numpy as np
import pandas as pd
from scipy.stats import norm
import plotly.express as px
import plotly.io as pio
pio.renderers.default='notebook'

# Parameters
mn1 = 0.15
sd1 = 0.10
mn2 = 0.35
sd2 = 0.10
T = 50

# Generate data
rv1   = norm(loc=mn1, scale=sd1).rvs(T)
rv2   = norm(loc=mn2, scale=sd2).rvs(T)
time  = np.arange(T)
df = pd.DataFrame(data={'time': time, 'ret1': rv1, 'ret2': rv2})

# Plot data
fig = px.line(df,x='time', y=['ret1', 'ret2'])
fig.update_layout(yaxis_title='Return',
                  xaxis_title='',
    legend_title_text='',
)
fig.show()

Which return series do you prefer?

Code 
# Parameters
mn1 = 0.20
sd1 = 0.10
mn2 = 0.20
sd2 = 0.40
T = 50

# Generate data
rv1   = norm(loc=mn1, scale=sd1).rvs(T)
rv2   = norm(loc=mn2, scale=sd2).rvs(T)
time  = np.arange(T)
df = pd.DataFrame(data={'time': time, 'ret1': rv1, 'ret2': rv2})

# Plot data
fig = px.line(df,x='time', y=['ret1', 'ret2'])
fig.update_layout(yaxis_title='Return',
                  xaxis_title='',
    legend_title_text='',
)
fig.show()

Which return series do you prefer?

Code 
# Parameters
mn1 = 0.20
sd1 = 0.10
mn2 = 0.40
sd2 = 0.40
T = 50

# Generate data
rv1   = norm(loc=mn1, scale=sd1).rvs(T)
rv2   = norm(loc=mn2, scale=sd2).rvs(T)
time  = np.arange(T)
df = pd.DataFrame(data={'time': time, 'ret1': rv1, 'ret2': rv2})

# Plot data
fig = px.line(df,x='time', y=['ret1', 'ret2'])
fig.update_layout(yaxis_title='Return',
                  xaxis_title='',
    legend_title_text='',
)
fig.show()

Where would your portfolio like to live?

Code 
import matplotlib.pyplot as plt
fig, ax = plt.subplots()
ax.set_xlim([0,0.5])
ax.set_ylim([0,0.25])
ax.set_ylabel(r'$E[r_p]$')
ax.set_xlabel(r'$\[r_p]$')

# mn = np.arange(0,0.5,0.1)
# sd = np.arange(0,0.5,0.1)

# ax.plot(sd, sd*0, color='k')
# ax.plot(mn*0,mn, color='k')
Text(0.5, 0, '$\\text{sd}[r_p]$')
ValueError: 
\text{sd}[r_p]
^
Unknown symbol: \text, found '\'  (at char 0), (line:1, col:1)
<Figure size 432x288 with 1 Axes>

BUSI 448